VALUES OF THE EULER φ-FUNCTION NOT DIVISIBLE BY A GIVEN ODD PRIME, AND THE DISTRIBUTION OF EULER-KRONECKER CONSTANTS FOR CYCLOTOMIC FIELDS

摘要

Let φ denote Euler’s phi function. For a fixed odd prime q we investigate the first and second order terms of the asymptotic series expansion for the number of n ≤ x such that q | φ(n). Part of the analysis involves a careful study of the Euler-Kronecker constants for cyclotomic fields. In particular, we show that the Hardy-Littlewood conjecture about counts of prime k-tuples and a conjecture of Ihara about the distribution of these Euler- Kronecker constants cannot be both true.